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Foliation of an asymptotically flat end by critical capacitors

Mouhammed Moustapha Fall, Ignace Aristide Minlend, Jesse Ratzkin

TL;DR

The paper addresses foliation of an asymptotically flat end by hypersurfaces that are critical points of a capacity-based functional, translating a potential-theoretic over-determined boundary value problem into a geometric foliation question. The authors reformulate the problem on a fixed domain using a large radius parameter \\rho, a translation \\tau, and a boundary deformation \\w, then construct an approximate harmonic solution and correct it via a nonlinear analysis that hinges on inverting the Dirichlet-to-Neumann map for the Laplace-Beltrami operator. A detailed linear analysis identifies the role of a kernel corresponding to rigid motions and establishes invertibility on the orthogonal complement, which is then exploited with the implicit function theorem to solve for \\tau(\\rho,\\w) and \\w(\\rho). The result is a smooth one-parameter foliation \\partial \\Omega_{\\rho} of the AF end by extremal capacitors with explicit Neumann data, generalizing classical foliations such as Huisken–Yau to a capacity setting and linking potential theory with geometric structure in general relativity contexts.

Abstract

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace-Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the non-local nature of our problem

Foliation of an asymptotically flat end by critical capacitors

TL;DR

The paper addresses foliation of an asymptotically flat end by hypersurfaces that are critical points of a capacity-based functional, translating a potential-theoretic over-determined boundary value problem into a geometric foliation question. The authors reformulate the problem on a fixed domain using a large radius parameter \\rho, a translation \\tau, and a boundary deformation \\w, then construct an approximate harmonic solution and correct it via a nonlinear analysis that hinges on inverting the Dirichlet-to-Neumann map for the Laplace-Beltrami operator. A detailed linear analysis identifies the role of a kernel corresponding to rigid motions and establishes invertibility on the orthogonal complement, which is then exploited with the implicit function theorem to solve for \\tau(\\rho,\\w) and \\w(\\rho). The result is a smooth one-parameter foliation \\partial \\Omega_{\\rho} of the AF end by extremal capacitors with explicit Neumann data, generalizing classical foliations such as Huisken–Yau to a capacity setting and linking potential theory with geometric structure in general relativity contexts.

Abstract

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace-Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the non-local nature of our problem

Paper Structure

This paper contains 14 sections, 14 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Reformulation of the problem
  3. Preliminary computations
  4. Notation
  5. Metric expansions and the Laplacian
  6. Weighted spaces
  7. Approximate and actual solutions
  8. Expansion of the normal derivative and linear analysis
  9. Expansion of the normal derivative
  10. Linear analysis of the normal derivative
  11. The spectral properties of the operator $\mathbb{L}$
  12. Solving the nonlinear problem
  13. Foliation by boundaries of extremal capacitors
  14. Variational setting

Key Result

Theorem 1

Let $(M,g)$ be a Riemannian manifold of dimension $n\geq 3$ with one asymptotically flat end $M \backslash K$, parameterized as in end-parameterization1 and metric-expansion1. Then there exists $\rho_0>1$ and compact sets $K_\rho$ indexed by $\rho \in (\rho_0, \infty)$ such that the domains $\Omega_ where $\eta$ is the unit interior normal to $K_\rho$ and The hypersurfaces $\{ \partial K_\rho\}_{

Theorems & Definitions (27)

  • Theorem 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Lemma 3.3
  • ...and 17 more